I learned the following from Constantin Teleman, and from these lectures of David Ben-Zvi.
Let $G$ be a finite group. Let $A = \mathbb{C}[G]^G$. This is an algebra under convolution:
$$(\phi \cdot \psi)(g) := \sum_{h \in G}\phi(gh^{-1})\psi(h).$$
We also have a trace $t : A \to \mathbb{C}$ given by $t(\phi) = \phi(1)/|G|$.
This is a Frobenius algebra. (Maybe this is the original example of a Frobenius algebra?)
In fact, this is a semisimple Frobenius algebra: Let $P_i = \frac{\chi_i \operatorname{dim}\chi_i}{|G|}$, where $\chi_i$ are the irreducible characters. Then $P_i$ is a basis for $A$ and $P_i\cdot P_j = \delta_{ij}P_i$.
Note that $1 = \sum_i P_i$. We have $t(1) = t(\sum_i P_i) = \frac{\sum_i (\operatorname{dim} \chi_i)^2}{|G|^2}$. On the other hand we have $t(1) = 1/|G|$. We get the formula $$|G| = \sum_i (\operatorname{dim} \chi_i)^2.$$
Pretty cool, no?
I think the correspondence between Frobenius algebras and 2D TQFTs is more than just "teaching an old dog new tricks". The 2D TQFT corresponding to this Frobenius algebra is the "finite group ('pure') gauge theory" of Dijkgraaf-Witten. From the TQFT perspective, $t(1)$ is the number which is assigned to the sphere $S^2$. It's some kind of "path integral"(?) over all maps $S^2 \to BG = \ast/G$. This number is a certain "weighted count" of $G$-bundles over $S^2$. And of course we have a similar story for all other surfaces. When the surfaces have boundary, we look at $G$-bundles with prescribed holonomy around the boundary circles.
Moreover, the 2D TQFT can be "extended" to manifolds with corners, which is also interesting. Check out Ben-Zvi's lecture notes for more on this stuff.
